Projective line over the finite quotient ring GF(2)[x]/〈x3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2007-05

AUTHORS

M. Saniga, M. Planat, M. Minarovjech

ABSTRACT

In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres “magic” square and the Mermin pentagram. The former is a 3×3 array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2) ⊗ GF(2). Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column’s observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over GF(2)[x]/〈x3 ™ x〉. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over GF(2), their three “Jacobson” counterparts, and the four points whose both coordinates are zero divisors. The other con.guration features the neighborhood of the point (1, 0) (or, equivalently, that of (0, 1)). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions. More... »

PAGES

625-631

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11232-007-0049-5

DOI

http://dx.doi.org/10.1007/s11232-007-0049-5

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1028110868


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/1608", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Sociology", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/16", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Studies in Human Society", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Astronomical Institute", 
          "id": "https://www.grid.ac/institutes/grid.493212.f", 
          "name": [
            "Astronomical Institute, Slovak Academy of Sciences, Tatransk\u00e1 Lomnica, Slovak Republic"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Saniga", 
        "givenName": "M.", 
        "id": "sg:person.015610617470.96", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015610617470.96"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Franche Comt\u00e9 \u00c9lectronique M\u00e9canique Thermique et Optique Sciences et Technologies", 
          "id": "https://www.grid.ac/institutes/grid.462068.e", 
          "name": [
            "D\u00e9partement LPMO, Institut FEMTO-ST, CNRS, Besan\u00e7on, France"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Planat", 
        "givenName": "M.", 
        "id": "sg:person.016076407625.27", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016076407625.27"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Astronomical Institute", 
          "id": "https://www.grid.ac/institutes/grid.493212.f", 
          "name": [
            "Astronomical Institute, Slovak Academy of Sciences, Tatransk\u00e1 Lomnica, Slovak Republic"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Minarovjech", 
        "givenName": "M.", 
        "id": "sg:person.010346430401.17", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010346430401.17"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "https://doi.org/10.1103/physrev.47.777", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1009714864"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/physrev.47.777", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1009714864"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/s11232-007-0035-y", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1015916706", 
          "https://doi.org/10.1007/s11232-007-0035-y"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/s11232-007-0035-y", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1015916706", 
          "https://doi.org/10.1007/s11232-007-0035-y"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1023/a:1007863413622", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1025782171", 
          "https://doi.org/10.1023/a:1007863413622"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1088/0305-4470/39/2/013", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1028113054"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1088/0305-4470/39/2/013", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1028113054"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1016/0039-3681(93)90061-n", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1040262349"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1088/0305-4470/24/4/003", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1059072003"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/physrevlett.65.3373", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1060801792"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/physrevlett.65.3373", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1060801792"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/revmodphys.38.447", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1060838481"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/revmodphys.38.447", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1060838481"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/revmodphys.65.803", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1060839301"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1103/revmodphys.65.803", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1060839301"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1119/1.1773173", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1062236865"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1142/s0217979206034388", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1062934750"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1512/iumj.1968.17.17004", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1067510923"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "2007-05", 
    "datePublishedReg": "2007-05-01", 
    "description": "In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres \u201cmagic\u201d square and the Mermin pentagram. The former is a 3\u00d73 array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2) \u2297 GF(2). Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column\u2019s observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over GF(2)[x]/\u3008x3 \u2122 x\u3009. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over GF(2), their three \u201cJacobson\u201d counterparts, and the four points whose both coordinates are zero divisors. The other con.guration features the neighborhood of the point (1, 0) (or, equivalently, that of (0, 1)). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions.", 
    "genre": "research_article", 
    "id": "sg:pub.10.1007/s11232-007-0049-5", 
    "inLanguage": [
      "en"
    ], 
    "isAccessibleForFree": true, 
    "isPartOf": [
      {
        "id": "sg:journal.1327888", 
        "issn": [
          "0040-5779", 
          "2305-3135"
        ], 
        "name": "Theoretical and Mathematical Physics", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "2", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "151"
      }
    ], 
    "name": "Projective line over the finite quotient ring GF(2)[x]/\u3008x3 \u2122 x\u3009 and quantum entanglement: The Mermin \u201cmagic\u201d square/pentagram", 
    "pagination": "625-631", 
    "productId": [
      {
        "name": "readcube_id", 
        "type": "PropertyValue", 
        "value": [
          "85e9b811902b5ee756efd59660b95ecbbda81d37bde32ae8d62b394c165e2772"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/s11232-007-0049-5"
        ]
      }, 
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1028110868"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/s11232-007-0049-5", 
      "https://app.dimensions.ai/details/publication/pub.1028110868"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2019-04-10T21:39", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8687_00000522.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "http://link.springer.com/10.1007%2Fs11232-007-0049-5"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s11232-007-0049-5'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s11232-007-0049-5'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s11232-007-0049-5'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s11232-007-0049-5'


 

This table displays all metadata directly associated to this object as RDF triples.

116 TRIPLES      21 PREDICATES      39 URIs      19 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/s11232-007-0049-5 schema:about anzsrc-for:16
2 anzsrc-for:1608
3 schema:author Ne4af3e540de6470e9d5aa87ee4b213b3
4 schema:citation sg:pub.10.1007/s11232-007-0035-y
5 sg:pub.10.1023/a:1007863413622
6 https://doi.org/10.1016/0039-3681(93)90061-n
7 https://doi.org/10.1088/0305-4470/24/4/003
8 https://doi.org/10.1088/0305-4470/39/2/013
9 https://doi.org/10.1103/physrev.47.777
10 https://doi.org/10.1103/physrevlett.65.3373
11 https://doi.org/10.1103/revmodphys.38.447
12 https://doi.org/10.1103/revmodphys.65.803
13 https://doi.org/10.1119/1.1773173
14 https://doi.org/10.1142/s0217979206034388
15 https://doi.org/10.1512/iumj.1968.17.17004
16 schema:datePublished 2007-05
17 schema:datePublishedReg 2007-05-01
18 schema:description In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres “magic” square and the Mermin pentagram. The former is a 3×3 array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2) ⊗ GF(2). Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column’s observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over GF(2)[x]/〈x3 ™ x〉. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over GF(2), their three “Jacobson” counterparts, and the four points whose both coordinates are zero divisors. The other con.guration features the neighborhood of the point (1, 0) (or, equivalently, that of (0, 1)). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions.
19 schema:genre research_article
20 schema:inLanguage en
21 schema:isAccessibleForFree true
22 schema:isPartOf N1347e9a83e8543f897215757231c490d
23 N1cde95936ce1484596b2eb8dd0d4af39
24 sg:journal.1327888
25 schema:name Projective line over the finite quotient ring GF(2)[x]/〈x3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram
26 schema:pagination 625-631
27 schema:productId Nb9c7f945f6a64a55890d419661e4efb4
28 Nf53597377da240b5bff96fcc77deaa01
29 Nfc444bfc9eda47dbabe2b9816b664948
30 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028110868
31 https://doi.org/10.1007/s11232-007-0049-5
32 schema:sdDatePublished 2019-04-10T21:39
33 schema:sdLicense https://scigraph.springernature.com/explorer/license/
34 schema:sdPublisher Ned65bb4bc34945ab83561a5def8da785
35 schema:url http://link.springer.com/10.1007%2Fs11232-007-0049-5
36 sgo:license sg:explorer/license/
37 sgo:sdDataset articles
38 rdf:type schema:ScholarlyArticle
39 N1347e9a83e8543f897215757231c490d schema:issueNumber 2
40 rdf:type schema:PublicationIssue
41 N1cde95936ce1484596b2eb8dd0d4af39 schema:volumeNumber 151
42 rdf:type schema:PublicationVolume
43 N329d7d2bfe664ebd863dda039c76bd4f rdf:first sg:person.016076407625.27
44 rdf:rest N57f9ff5b257b4e34a2aaf6995a8c35be
45 N57f9ff5b257b4e34a2aaf6995a8c35be rdf:first sg:person.010346430401.17
46 rdf:rest rdf:nil
47 Nb9c7f945f6a64a55890d419661e4efb4 schema:name readcube_id
48 schema:value 85e9b811902b5ee756efd59660b95ecbbda81d37bde32ae8d62b394c165e2772
49 rdf:type schema:PropertyValue
50 Ne4af3e540de6470e9d5aa87ee4b213b3 rdf:first sg:person.015610617470.96
51 rdf:rest N329d7d2bfe664ebd863dda039c76bd4f
52 Ned65bb4bc34945ab83561a5def8da785 schema:name Springer Nature - SN SciGraph project
53 rdf:type schema:Organization
54 Nf53597377da240b5bff96fcc77deaa01 schema:name dimensions_id
55 schema:value pub.1028110868
56 rdf:type schema:PropertyValue
57 Nfc444bfc9eda47dbabe2b9816b664948 schema:name doi
58 schema:value 10.1007/s11232-007-0049-5
59 rdf:type schema:PropertyValue
60 anzsrc-for:16 schema:inDefinedTermSet anzsrc-for:
61 schema:name Studies in Human Society
62 rdf:type schema:DefinedTerm
63 anzsrc-for:1608 schema:inDefinedTermSet anzsrc-for:
64 schema:name Sociology
65 rdf:type schema:DefinedTerm
66 sg:journal.1327888 schema:issn 0040-5779
67 2305-3135
68 schema:name Theoretical and Mathematical Physics
69 rdf:type schema:Periodical
70 sg:person.010346430401.17 schema:affiliation https://www.grid.ac/institutes/grid.493212.f
71 schema:familyName Minarovjech
72 schema:givenName M.
73 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010346430401.17
74 rdf:type schema:Person
75 sg:person.015610617470.96 schema:affiliation https://www.grid.ac/institutes/grid.493212.f
76 schema:familyName Saniga
77 schema:givenName M.
78 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015610617470.96
79 rdf:type schema:Person
80 sg:person.016076407625.27 schema:affiliation https://www.grid.ac/institutes/grid.462068.e
81 schema:familyName Planat
82 schema:givenName M.
83 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016076407625.27
84 rdf:type schema:Person
85 sg:pub.10.1007/s11232-007-0035-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1015916706
86 https://doi.org/10.1007/s11232-007-0035-y
87 rdf:type schema:CreativeWork
88 sg:pub.10.1023/a:1007863413622 schema:sameAs https://app.dimensions.ai/details/publication/pub.1025782171
89 https://doi.org/10.1023/a:1007863413622
90 rdf:type schema:CreativeWork
91 https://doi.org/10.1016/0039-3681(93)90061-n schema:sameAs https://app.dimensions.ai/details/publication/pub.1040262349
92 rdf:type schema:CreativeWork
93 https://doi.org/10.1088/0305-4470/24/4/003 schema:sameAs https://app.dimensions.ai/details/publication/pub.1059072003
94 rdf:type schema:CreativeWork
95 https://doi.org/10.1088/0305-4470/39/2/013 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028113054
96 rdf:type schema:CreativeWork
97 https://doi.org/10.1103/physrev.47.777 schema:sameAs https://app.dimensions.ai/details/publication/pub.1009714864
98 rdf:type schema:CreativeWork
99 https://doi.org/10.1103/physrevlett.65.3373 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060801792
100 rdf:type schema:CreativeWork
101 https://doi.org/10.1103/revmodphys.38.447 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060838481
102 rdf:type schema:CreativeWork
103 https://doi.org/10.1103/revmodphys.65.803 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060839301
104 rdf:type schema:CreativeWork
105 https://doi.org/10.1119/1.1773173 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062236865
106 rdf:type schema:CreativeWork
107 https://doi.org/10.1142/s0217979206034388 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062934750
108 rdf:type schema:CreativeWork
109 https://doi.org/10.1512/iumj.1968.17.17004 schema:sameAs https://app.dimensions.ai/details/publication/pub.1067510923
110 rdf:type schema:CreativeWork
111 https://www.grid.ac/institutes/grid.462068.e schema:alternateName Franche Comté Électronique Mécanique Thermique et Optique Sciences et Technologies
112 schema:name Département LPMO, Institut FEMTO-ST, CNRS, Besançon, France
113 rdf:type schema:Organization
114 https://www.grid.ac/institutes/grid.493212.f schema:alternateName Astronomical Institute
115 schema:name Astronomical Institute, Slovak Academy of Sciences, Tatranská Lomnica, Slovak Republic
116 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...